This summer, John Oliver, “Senior British Correspondent” on “The Daily Show,” will hijack the anchor’s chair from Jon Stewart. Photo: Contour by Getty Images

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Comedian John Oliver runs out onstage with a big smile, waving to a packed house at Webster Hall on Tuesday night. He is taping a stand-up set for the fourth season of his Comedy Central series “John Oliver’s New York Stand-Up Show.”

While a stylist adjusts his tie and last-minute tweaks are made to the lighting, Oliver asks the crowd to laugh loudly — “I realize how desperate it sounds for me, as a comedian, to ask you to laugh at my jokes,” he says.

Someone in the crowd asks if the comedian is excited to fill in for his boss this summer on “The Daily Show With Jon Stewart.”

“The only thing I’m nervous about is talking to guests like human beings, because all of my interviews so far have been attacking people,” Oliver says. “I have a genuine concern about sitting across from an actor whose movies I obviously haven’t seen.

“Two nights ago, I had a nightmare I was interviewing Jaden Smith. I said, ‘Look, Jaden, I haven’t seen “After Earth” but the trailers look awful,’ and behind him, I could see Will Smith doing this [raises a fist]. I don’t know why in my head that’s how Will Smith expresses anger, like a 1920s journalist.”

Oliver, 36, has made a name for himself taking digs at politicians and other partisans in edited segments as Stewart’s “Senior British Correspondent” since 2006.

The dark-haired, bespectacled Brit has a self-deprecating, satiric sense of humor, which has earned him a cult following in his seven years on “The Daily Show.” He has the ninth most popular comedy podcast on iTunes, more than 14,000 Facebook likes and nearly 90,000 Twitter followers, even though he rarely tweets.

But he has no experience sitting in the swivel chair, interviewing celebs or doing the monologue — all skills integral to the high-profile gig.

Oliver will start his three-month stint filling in on Monday, while Stewart takes the summer off to direct his first movie, “Rosewater.”

It’s a huge opportunity for Oliver. The Emmy-winning news satire show regularly ranks No. 1 in the coveted 18 to 49 demo for late night, beating both “Late Night With David Letterman” and “The Tonight Show With Jay Leno.” The show averaged 2.5 million total viewers in the first quarter of this year.

While fans of Stewart champion Oliver, this hosting gig will be his chance to test his mass appeal and ability to anchor his own show.

Stewart has employed fill-ins a few other times since taking over the hosting gig from Craig Kilborn in 1999. Rob Corddry, Stephen Colbert and Steve Carell sat in for different stints when Stewart’s two children were born. But this summer will be Stewart’s longest absence from the show.

And it’s a long way from Oliver’s early days touring student unions with Andy Zaltzman, his current co-host for the popular comedy podcast “The Bugle.”

“[Hosting ‘The Daily Show’] is obviously a big opportunity,” Zaltzman tells The Post. “It will certainly seem like he has come a long way since we were standing onstage in an empty theater in East London at an Edinburgh [Festival] preview show in 2004 when our entire audience — of four people — stood up and left after 20 minutes.”

At that point, Oliver had never even been to America. A couple of years later, he would make his first trip across the pond after fellow Brit Ricky Gervais told Stewart about his talents. He had seen Oliver perform stand-up and was impressed.

“I came over and pretty much started the next day,” Oliver told the Independent in a 2010 interview. “I had never been to America before, but I guess I had a workable knowledge, just because that’s necessary to know how the world works . . . but in terms of the intricacies of how America was governed, I certainly did not know a lot, so that was kind of a crash course.”

Oliver had a work visa for the first couple of years here, and then got a green card in late 2009.

Two years later, Oliver married an American, Kate Norley. The couple met at the 2008 Republican National Convention — Oliver was covering the event for “The Daily Show” and Norley, a former Iraq War combat medic, was there on behalf of a veterans’ rights group.

“It’s the most emasculating thing I could possibly do to go out with someone who has actually done something valuable with their life,” Oliver joked to People magazine when the couple announced their engagement in 2010.

Oliver was born in Birmingham and went to school at Cambridge University, where he studied English and performed with the comedy troupe the Footlights. His first stand-up show was in 2002 at the Edinburgh Festival.

He stays true to his roots these days by cheering on Liverpool, the English Premier League soccer team, but he has taken a passionate liking to the New York Mets, like his boss, Stewart.

Oliver will likely play up his British background while filling in for Stewart, who is a native New Yorker, says Brad Adgate, director of research at Horizon Media.

Zaltzman, who still lives in England, agrees.

“It will become a vehicle for rabidly pro-British propaganda,” he jokes. “He’ll probably demand the reintroduction of the Stamp Act, and by August, America will be a British colony again.”